Theorem 1.  If the complex functions  $f_{1},\,f_{2},\,f_{3},\,\ldots$  are continuous on the path $\gamma$ and the series

converges uniformly on $\gamma$ to the sum function $F$, then one has

Theorem 2.  If the functions  $f_{1},\,f_{2},\,f_{3},\,\ldots$  are holomorphic in a domain $A$ and the series (1) converges uniformly in every closed disc of $A$, then also the sum function $F$ of (1) is holomorphic in $A$ and the equality

is true for every positive integer $n$ in all points of $A$.  The series (2) converges uniformly in every compact subdomain of $A$.

Theorem 3.  If $f(z)$ is holomorphic in a domain $A$ and $z_{0}$ is a point of $A$, then one can expand $f(z)$ to a power series (the so-called Taylor series)

This expansion is valid at least in the greatest disk  $|z-z_{0}|<r\;(\leqq\infty)$  which contains points of $A$ only.